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8/3/2019 Specification Test
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Specification test
Vid Adrison
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Outline
Redundant Variable
Omitted Variable
Functional Specification
Selection Criteria
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Redundant Variable Consequences
On the unbiasedness: remain unbiased Review the concept of unbiased estimator
On the variance: increases variance Proof:
Create a simulated demand function Simulation is useful as we know the true value of the parameter
Steps in conducting simulation; Assume that Qx is only a function of Px and Income Generate 200 data of Px, Py, INC, and Error via random draw
In excel the syntax is =rand() Generate log(Qx)= 0.5-0.5*log(Px)+0.5*log(INC)+Error Run log(Qx)=f[log(Px), log(INC)]
The parameter will be closer to the assigned values, as thenumber of draws increase
Repeating the above procedure for N times and get the averagevalues of the parameter Monte Carlo Simulation
As the comparison, run log(Qx)=f[log(Px), log(Py), log(INC)], seehow the parameter changes
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Redundant Variable
Test procedure in EVIEWS:
View | Coefficient Test | Omitted Variables | (WriteVariables | OK
H0: Variables do not belong to the model
H1: Variables belong to the model
This procedure is the same as omitted variabletest, thus, the hypotheses remain the same
Basically, omitted variable/redundant variable testare performed by comparing the likelihood ratiobetween restricted and unrestricted model
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Correct Specification RegressionDependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:44Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
LOG(PX) -0.525034 0.035679 -14.71562 0.0000LOG(INC) 0.514221 0.045908 11.20119 0.0000
C 0.970042 0.095809 10.12477 0.0000
R-squared 0.828189 Mean dependent var 1.237024Adjusted R-squared 0.822161 S.D. dependent var 0.723513S.E. of regression 0.305112 Akaike info criterion 0.512434Sum squared resid 5.306335 Schwarz criterion 0.617151Log likelihood -12.37302 F-statistic 137.3802Durbin-Watson stat 2.276588 Prob(F-statistic) 0.000000
Redundant Variable caseDependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:44Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
LOG(PX) -0.521201 0.035292 -14.76838 0.0000LOG(INC) 0.528201 0.046149 11.44567 0.0000LOG(PY) 0.070505 0.044328 1.590528 0.1173
C 0.890289 0.107022 8.318742 0.0000
R-squared 0.835615 Mean dependent var 1.237024Adjusted R-squared 0.826809 S.D. dependent var 0.723513S.E. of regression 0.301099 Akaike info criterion 0.501583Sum squared resid 5.076984 Schwarz criterion 0.641206Log likelihood -11.04750 F-statistic 94.88810Durbin-Watson stat 2.360587 Prob(F-statistic) 0.000000
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Omitted Variable Consequences
On the unbiasedness: more serious than redundantvariable case Omitted variable may be due to ignorance or data
unavailability
Example:
Dropping INC from the previous regression Excluding ability in wage offer function
For two variable-model, the sign of bias depends on thecorrelation between excluded variable and includedvariable
The direction of bias can be more complicated if we havethree or more regressors
See Wooldridge Chapter 3 for detail derivation
Corr (X1, X2) > 0 Corr(X1, X2) 0 Positive Bias Negative Bias
B2 < 0 Negative Bias Positive Bias
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Omitted Variable case
Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:45Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
LOG(PX) -0.420876 0.061096 -6.888789 0.0000C 1.800707 0.107595 16.73598 0.0000
R-squared 0.450005 Mean dependent var 1.237024Adjusted R-squared 0.440522 S.D. dependent var 0.723513S.E. of regression 0.541175 Akaike info criterion 1.642617Sum squared resid 16.98648 Schwarz criterion 1.712429Log likelihood -47.27851 F-statistic 47.45541Durbin-Watson stat 1.828653 Prob(F-statistic) 0.000000
Omitted Variable Test
Omitted Variables: LOG(INC)
F-statistic 125.4667 Probability 0.000000Log likelihood ratio 69.81100 Probability 0.000000
Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 23:52
Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
C 0.970042 0.095809 10.12477 0.0000LOG(PX) -0.525034 0.035679 -14.71562 0.0000LOG(INC) 0.514221 0.045908 11.20119 0.0000
R-squared 0.828189 Mean dependent var 1.237024Adjusted R-squared 0.822161 S.D. dependent var 0.723513S.E. of regression 0.305112 Akaike info criterion 0.512434Sum squared resid 5.306335 Schwarz criterion 0.617151
Log likelihood -12.37302 F-statistic 137.3802Durbin-Watson stat 2.276588 Prob(F-statistic) 0.000000
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Regression through Origin
Recall the interpretation of intercept
For Keynesian consumption function, it reflectsautonomous consumption; the amount of consumptionone will have if his/her income is zero
Some have no (logical) economic interpretation: I.e., production function (K=0, L=0 will definitely lead to
Y=0, demand function (price should be in the positivedomain)
In the absence of economic interpretation, one is
tempted to drop intercept It is essentially dropping vector of ONE in the matrix
notation
Is it the correct treatment ???
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Regression through Origin Note that an intercept does not have to have economic
interpretation One of several role of an intercept is to ensure zero conditional
mean on error Example of violation;
True Consumption = B0 + B1*Income + error If consumption is measured incorrectly, such as, understatement;
such that Observed consumption = True consumption understatement The regression would be;
Observed Consumption = B0 + B1*Income + error understatement
If we dont include B0, then E (error understatement) is differentfrom zero Bias in B1
If we include B0, B1 is not biased
Cost of using intercept if B0 is truly zero None Cost of deleting intercept if B0 is not zero Biased in slope
parameter
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Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 18:18
Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
LOG(PX) -0.429977 0.057083 -7.532469 0.0000LOG(INC) 0.873992 0.048203 18.13144 0.0000
R-squared 0.519198 Mean dependent var 1.237024Adjusted R-squared 0.510909 S.D. dependent var 0.723513S.E. of regression 0.505989 Akaike info criterion 1.508162Sum squared resid 14.84945 Schwarz criterion 1.577973Log likelihood -43.24485 Durbin-Watson stat 1.730641
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Functional Specification What to choose:
A: ln(Qx)=f(Px, INC), B: ln(Qx)=f(Px, Py, INC), C: ln(Qx)=f(ln(Px),ln(INC)) D: ln(Qx)=f(ln(Px),ln(Py), ln(INC))??
Nested Model: A Vs B, or C Vs D Ramsey RESET
Basically add the polynomial of expected value as theregressor, as the proxy for unaccounted variable
If adding this proxy variable leads to significant increase inadjusted R square, the regression contains misspecification
Steps in Eviews: View | Stability Test | Ramsey RESETtest | (Include number of polynomial variable) | OK
H0: No misspecification error H1: Model contains specification error
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Ramsey RESET Test:
F-statistic 0.784074 Probability 0.379684Log likelihood ratio 0.834253 Probability 0.361046
Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:33Sample: 1 60
Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.
C 1.088893 0.165015 6.598762 0.0000LOG(PX) -0.601195 0.093144 -6.454500 0.0000LOG(INC) 0.550684 0.061735 8.920113 0.0000FITTED^2 -0.043772 0.049433 -0.885480 0.3797
R-squared 0.830562 Mean dependent var 1.237024Adjusted R-squared 0.821485 S.D. dependent var 0.723513S.E. of regression 0.305692 Akaike info criterion 0.531863Sum squared resid 5.233065 Schwarz criterion 0.671486Log likelihood -11.95589 F-statistic 91.50122
Durbin-Watson stat 2.255349 Prob(F-statistic) 0.000000
Ramsey RESET Test:
F-statistic 4.159492 Probability 0.046131Log likelihood ratio 4.298853 Probability 0.038138
Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:34Sample: 1 60
Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.
C 0.502144 0.492582 1.019413 0.3124PX 0.005691 0.080800 0.070429 0.9441INC -0.004399 0.042664 -0.103115 0.9182
FITTED^2 0.413044 0.202524 2.039483 0.0461
R-squared 0.543989 Mean dependent var 1.237024Adjusted R-squared 0.519560 S.D. dependent var 0.723513S.E. of regression 0.501494 Akaike info criterion 1.521890Sum squared resid 14.08379 Schwarz criterion 1.661513Log likelihood -41.65671 F-statistic 22.26804Durbin-Watson stat 2.432008 Prob(F-statistic) 0.000000
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Functional Specification
Non Nested Model: A Vs C (In theprevious slides)
Mizon and Richard (1986)
Ln(Qx) =B0 + B1*Px +B2*INC+B3*ln(Px)+B4*ln(INC)+e
Test using Wald
B1=B2=0 if null is rejected, then specification A ispreferred
B3=B4=0 if null is rejected, then specification C ispreferred
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Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:55Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.C 1.047136 0.119279 8.778901 0.0000
LOG(PX) -0.490712 0.064609 -7.595069 0.0000LOG(INC) 0.604477 0.083606 7.230085 0.0000
PX -0.017596 0.025031 -0.702977 0.4850INC -0.024155 0.018644 -1.295620 0.2005
R-squared 0.834092 Mean dependent var 1.237024Adjusted R-squared 0.822026 S.D. dependent var 0.723513S.E. of regression 0.305228 Akaike info criterion 0.544139Sum squared resid 5.124023 Schwarz criterion 0.718668
Log likelihood -11.32417 F-statistic 69.12739Durbin-Watson stat 2.314130 Prob(F-statistic) 0.000000
Wald Test:Equation: Untitled
Null Hypothesis: C(4)=0C(5)=0
F-statistic 0.978447 Probability 0.382341Chi-square 1.956895 Probability 0.375894
Wald Test:Equation: Untitled
Null Hypothesis: C(2)=0C(3)=0
F-statistic 53.70023 Probability 0.000000Chi-square 107.4005 Probability 0.000000
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Functional Specification
Davidson-MacKinnon (1981) Use the similar principle as Ramsey, but different
predicted values
Recall Spec A: ln(Qx)=f(Px, INC)
Spec C: ln(Qx)=f(ln(Px), ln(INC))
Steps: to test if Spec A is correct: Run Spec C, get predicted value, say Z1
Run Spec A by adding Z1 into the equation
If Z1 is insignificant, then A is correctly specified
We can also perform the test in the other direction; Run Spec A, get predicted value, say Z2
Run Spec C by adding Z2 into the equation
If Z2 is insignificant, then C is correctly specified
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Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:59Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
C 0.038918 0.176320 0.220726 0.8261PX 0.000279 0.020148 0.013860 0.9890INC -0.008157 0.013027 -0.626168 0.5337Z1 1.021596 0.099618 10.25511 0.0000
R-squared 0.829783 Mean dependent var 1.237024Adjusted R-squared 0.820665 S.D. dependent var 0.723513S.E. of regression 0.306393 Akaike info criterion 0.536446Sum squared resid 5.257103 Schwarz criterion 0.676069Log likelihood -12.09338 F-statistic 90.99747Durbin-Watson stat 2.239382 Prob(F-statistic) 0.000000
Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 01:01Sample: 1 60Included observations: 60
Variable Coefficient Std. Error t-Statistic Prob.
C 1.001958 0.176638 5.672389 0.0000LOG(PX) -0.534508 0.056757 -9.417454 0.0000
LOG(INC) 0.522164 0.059141 8.829176 0.0000Z2 -0.027657 0.128139 -0.215839 0.8299
R-squared 0.828332 Mean dependent var 1.237024Adjusted R-squared 0.819136 S.D. dependent var 0.723513S.E. of regression 0.307697 Akaike info criterion 0.544936Sum squared resid 5.301924 Schwarz criterion 0.684559Log likelihood -12.34807 F-statistic 90.07041Durbin-Watson stat 2.265905 Prob(F-statistic) 0.000000
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Selection Criteria
According to Hendry and Richard (1983), a modelchosen for empirical analysis should satisfy thefollowing criteria: Admissible (prediction made from the model must be
logically possible)
Consistent with theory: Make economic good sense Have weakly exogenous explanatory variables:
Regressors are uncorrelated with the error terms Constancy: The values of the parameters should be
stable. In other word, the parameter values obtainedusing within sample observation should not deviate
significantly from outside sample observation. Coherency: Residuals estimated from the model must be
purely random Encompassing: No other model explains better
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Selection Criteria
Evaluation of Competing Models
Three statistics for model evaluation criteriaavailable in most econometric software are; Adjusted R-Squared Choose model that generates
the highest Adjusted R squared
Akaike Information Criterion Choose model that
generates the smallest AIC
Schwarz Information Criterion Choose model that
generates the smallest SIC